IProve that exponential functions a n have different orders of growth for different values of base a&gt;0.It looks obvious that when a=3 it grows faster when compared to a=2. But how do i make a formal proof for this? Thanks for your help.

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IProve that exponential functions       a    n   have different orders of growth for different values of base a&amp;gt;0.It looks obvious that when a=3 it grows faster when compared to a=2. But how do i make a formal proof for this? Thanks for your help.

andytronicoh4t · Accepted Answer

Here&#039;s a sketch of a proof: suppose a,b&amp;gt;0, and   a  ≠  b. Without loss of generality, a&amp;gt;b. We want to show that   O  (      a    n    )  ≠  O  (      b    n    ), or equivalently, that       a    n    ∉  O  (      b    n    ) (why are these equivalent?)To show that       a    n    ∉  O  (      b    n    ), it suffices to show (again, why?) that      lim          n      →      ∞                  a      n              b      n        =  ∞Using the fact that a&amp;gt;b&amp;gt;0, this limit should be easy to show.

IProve that exponential functions an have different orders of growth for different values of base a>0. It looks obvious that when a=3 it grows faster when compared to a=2. But how do i make a formal proof for this? Thanks for your help.

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